Completeness Of Metric Space Research Articles

Cauchy Points of Metric Locales

A natural approach to topology which emphasizes its geometric essence independent of the notion of points is given by the concept of frame (for instance [4], [8]). We consider this a good formalization of the intuitive perception of a space as given by the “places” of non-trivial extent with appropriate geometric relations between them. Viewed from this position, points are artefacts determined by collections of places which may in some sense by considered as collapsing or contracting; the precise meaning of the latter as well as possible notions of equivalence being largely arbitrary, one may indeed have different notions of point on the same “space”. Of course, the well-known notion of a point as a homomorphism into 2 evidently fits into this pattern by the familiar correspondence between these and the completely prime filters. For frames equipped with a diameter as considered in this paper, we introduce a natural alternative, the Cauchy points. These are the obvious counterparts, for metric locales, of equivalence classes of Cauchy sequences familiar from the classical description of completion of metric spaces: indeed they are decreasing sequences for which the diameters tend to zero, identified by a natural equivalence relation.

Histoire des espaces complets

SUMMARY. — In the fifth century B.C., it is said, an unknown Pythagorean discovered the irrationality of √2. There begins the long history of the completion of the rationale, to which are tied the names of Euler, Bolzano, Cauchy, Méray, Heine, Cantor, Dedekind, and many others. The next chapter opens with Frécheťs generalization of the Cauchy criterion for metric spaces, Hausdorff's result on the completion of metric spaces, and the introduction of Banach spaces. This leads to the works of Weil, on complete uniform spaces, of Dieudonné, on topological spaces which can receive a complete uniform space structure, and of Grothendieck, on the completion of locally convex topological vector spaces.

Real-valued models with metric equality and uniformly continuous predicates

AbstractTwo real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.

A characterization of complete metric spaces

A general formulation of the completeness argument used in the Bishop-Phelps Theorem and many other places has been given by Ekeland. It is shown that Ekeland's formulation characterizes complete metric spaces. A central idea in the proof of the Bishop-Phelps Theorem is the use of norm completeness and a partial ordering to produce a point where a linear functional attains its supremum on a closed bounded convex set. In fact, this completeness technique is useful in many situations as has been described in the surveys of Phelps [7] and Brezis-Browder [1]. Recently Ekeland [3] has given a very general formulation of this technique and has applied it to a wide variety of problems [4]. In the present note we show that Ekeland's formulation is actually equivalent to completeness for metric spaces. Ekeland's Theorem may be stated as follows: THEOREM 1. Let (M, d) be a complete metric space, and F: M -R U { + oo} a lower semicontinuous function, F 5 + x, bounded from below. Let > 0 be given and a point u & M such that